Kraus

class Kraus(data, input_dims=None, output_dims=None)

Bases: qiskit.quantum_info.operators.channel.quantum_channel.QuantumChannel

Kraus representation of a quantum channel.

For a quantum channel $\mathcal{E}$ , the Kraus representation is given by a set of matrices $[A_0,...,A_{K-1}]$ such that the evolution of a DensityMatrix $\rho$ is given by

\mathcal{E}(\rho) = \sum_{i=0}^{K-1} A_i \rho A_i^\dagger

A general operator map $\mathcal{G}$ can also be written using the generalized Kraus representation which is given by two sets of matrices $[A_0,...,A_{K-1}]$ , $[B_0,...,A_{B-1}]$ such that

\mathcal{G}(\rho) = \sum_{i=0}^{K-1} A_i \rho B_i^\dagger

See reference [1] for further details.

References

C.J. Wood, J.D. Biamonte, D.G. Cory, Tensor networks and graphical calculus for open quantum systems, Quant. Inf. Comp. 15, 0579-0811 (2015). arXiv:1111.6950 [quant-ph]

Initialize a quantum channel Kraus operator.

Parameters

**(**QuantumCircuit or (data) – Instruction or BaseOperator or matrix): data to initialize superoperator.
input_dims (tuple) – the input subsystem dimensions. [Default: None]
output_dims (tuple) – the output subsystem dimensions. [Default: None]

Raises

QiskitError – if input data cannot be initialized as a a list of Kraus matrices.

Additional Information:

If the input or output dimensions are None, they will be automatically determined from the input data. If the input data is a list of Numpy arrays of shape (2**N, 2**N) qubit systems will be used. If the input does not correspond to an N-qubit channel, it will assign a single subsystem with dimension specified by the shape of the input.

Methods

adjoint

Kraus.adjoint()

Return the adjoint quantum channel.

Note

This is equivalent to the matrix Hermitian conjugate in the SuperOp representation ie. for a channel $\mathcal{E}$ , the SuperOp of the adjoint channel $\mathcal{{E}}^\dagger$ is $S_{\mathcal{E}^\dagger} = S_{\mathcal{E}}^\dagger$ .

compose

Kraus.compose(other, qargs=None, front=False)

Return the operator composition with another Kraus.

Parameters

other (Kraus) – a Kraus object.
qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).
front (bool) – If True compose using right operator multiplication, instead of left multiplication [default: False].

Returns

The composed Kraus.

Return type

Kraus

Raises

QiskitError – if other cannot be converted to an operator, or has incompatible dimensions for specified subsystems.

Note

Composition (&) by default is defined as left matrix multiplication for matrix operators, while dot() is defined as right matrix multiplication. That is that A & B == A.compose(B) is equivalent to B.dot(A) when A and B are of the same type.

Setting the front=True kwarg changes this to right matrix multiplication and is equivalent to the dot() method A.dot(B) == A.compose(B, front=True).

conjugate

Kraus.conjugate()

Return the conjugate quantum channel.

Note

This is equivalent to the matrix complex conjugate in the SuperOp representation ie. for a channel $\mathcal{E}$ , the SuperOp of the conjugate channel $\overline{{\mathcal{{E}}}}$ is $S_{\overline{\mathcal{E}^\dagger}} = \overline{S_{\mathcal{E}}}$ .

copy

Kraus.copy()

Make a deep copy of current operator.

dot

Kraus.dot(other, qargs=None)

Return the right multiplied operator self * other.

Parameters

other (Operator) – an operator object.
qargs (list or None) – Optional, a list of subsystem positions to apply other on. If None apply on all subsystems (default: None).

Returns

The right matrix multiplied Operator.

Return type

Operator

expand

Kraus.expand(other)

Return the reverse-order tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product $b \otimes a$ , where $a$

is the current Kraus, and $b$ is the other Kraus.

Return type

Kraus

input_dims

Kraus.input_dims(qargs=None)

Return tuple of input dimension for specified subsystems.

is_cp

Kraus.is_cp(atol=None, rtol=None)

Test if Choi-matrix is completely-positive (CP)

is_cptp

Kraus.is_cptp(atol=None, rtol=None)

Return True if completely-positive trace-preserving.

is_tp

Kraus.is_tp(atol=None, rtol=None)

Test if a channel is trace-preserving (TP)

is_unitary

Kraus.is_unitary(atol=None, rtol=None)

Return True if QuantumChannel is a unitary channel.

output_dims

Kraus.output_dims(qargs=None)

Return tuple of output dimension for specified subsystems.

power

Kraus.power(n)

Return the power of the quantum channel.

Parameters

n (float) – the power exponent.

Returns

the channel $\mathcal{{E}} ^n$ .

Return type

SuperOp

Raises

QiskitError – if the input and output dimensions of the SuperOp are not equal.

Note

For non-positive or non-integer exponents the power is defined as the matrix power of the SuperOp representation ie. for a channel $\mathcal{{E}}$ , the SuperOp of the powered channel $\mathcal{{E}}^n$ is $S_{{\mathcal{{E}}^n}} = S_{{\mathcal{{E}}}}^n$ .

reshape

Kraus.reshape(input_dims=None, output_dims=None, num_qubits=None)

Return a shallow copy with reshaped input and output subsystem dimensions.

Parameters

input_dims (None or tuple) – new subsystem input dimensions. If None the original input dims will be preserved [Default: None].
output_dims (None or tuple) – new subsystem output dimensions. If None the original output dims will be preserved [Default: None].
num_qubits (None or int) – reshape to an N-qubit operator [Default: None].

Returns

returns self with reshaped input and output dimensions.

Return type

BaseOperator

Raises

QiskitError – if combined size of all subsystem input dimension or subsystem output dimensions is not constant.

tensor

Kraus.tensor(other)

Return the tensor product with another Kraus.

Parameters

other (Kraus) – a Kraus object.

Returns

the tensor product $a \otimes b$ , where $a$

is the current Kraus, and $b$ is the other Kraus.

Return type

Kraus

Note

The tensor product can be obtained using the ^ binary operator. Hence a.tensor(b) is equivalent to a ^ b.

to_instruction

Kraus.to_instruction()

Convert to a Kraus or UnitaryGate circuit instruction.

If the channel is unitary it will be added as a unitary gate, otherwise it will be added as a kraus simulator instruction.

Returns

A kraus instruction for the channel.

Return type

qiskit.circuit.Instruction

Raises

QiskitError – if input data is not an N-qubit CPTP quantum channel.

to_operator

Kraus.to_operator()

Try to convert channel to a unitary representation Operator.

transpose

Kraus.transpose()

Return the transpose quantum channel.

Note

This is equivalent to the matrix transpose in the SuperOp representation, ie. for a channel $\mathcal{E}$ , the SuperOp of the transpose channel $\mathcal{{E}}^T$ is $S_{mathcal{E}^T} = S_{\mathcal{E}}^T$ .

Attributes

atol

Default value: 1e-08

data

Return list of Kraus matrices for channel.

dim

Return tuple (input_shape, output_shape).

num_qubits

Return the number of qubits if a N-qubit operator or None otherwise.

qargs

Return the qargs for the operator.

rtol

Default value: 1e-05

settings

Return settings.

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